Riemann Hypothesis

The hidden rhythm of the primes.

Last updated: 2025-10-25

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The Riemann zeta function ζ(s) encodes the distribution of prime numbers. The hypothesis states: all nontrivial zeros lie on the critical line Re(s)=1/2.

Proving it would sharpen prime-counting error terms and reverberate across number theory and cryptography.

Zeta and primes

For Re(s)>1: \(\zeta(s)=\sum_{n=1}^\infty n^{-s} = \prod_{p\ \text{prime}} (1-p^{-s})^{-1}\).

Nontrivial zeros \(\rho\) are conjectured to satisfy Re(\(\rho\))=1/2.

π(x)=Li(x)+O(x1/2log⁡x)(assuming RH)\pi(x) = \mathrm{Li}(x) + O\big(x^{1/2}\log x\big)\quad \text{(assuming RH)}
Zeros on the critical line: Im(s)=t vs. Z(t) sign changes.
Zeros on the critical line: Im(s)=t vs. Z(t) sign changes.

Why it matters

Sharper bounds influence cryptography (prime gaps, smooth numbers), random matrix theory links, and error terms in many arithmetic statistics.

Further reading